Overview
Title
The matrices with determinant forms the orthogonal group
O(n)={ A\in M_{n}(\mathbb{R}):A^{\top}A=I }
If we want to preserve orientation (let $\det A=1$) then we get the [[Special Orthogonal Group (Rotation)|special orthogonal group]] $SO(n)$ ## Relation to SO(n)O(n)/SO(n)={ \pm1 }
## Path-Connectedness From topological view, $SO(n)$ is [[Path Connectedness|path-connected]] while $O(n)$ is not. This is because the determinant cannot jump from $1$ to $-1$ along a continuous path (considering $\det A=1$ with $\det B=-1$, we cannot find a continuous path between them in $O(n)$)